Estimating functionals of one-dimensional Gibbs states

Ji, Chuanshu

doi:10.1007/bf00354757

Cited by 9 publications

(6 citation statements)

References 7 publications

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“…Higher order moments are obtained in the same way, especially second order moments related to the central limit theorem obeyed by Gibbs distributions (Bowen, 1975(Bowen, , 2008Keller, 1998). It is also possible to obtain averages of more complex functions than moments (Ji, 1989). The topological pressure is obtained via the spectrum of the Ruelle-Perron-Frobenius operator and can be calculated numerically when ψ has a finite (and small enough) range (see appendix for more details).…”

Section: Gibbs Measures As Statistical Modelsmentioning

confidence: 99%

How Gibbs Distributions May Naturally Arise from Synaptic Adaptation Mechanisms. A Model-Based Argumentation

et al. 2009

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This paper addresses two questions in the context of neuronal networks dynamics, using methods from dynamical systems theory and statistical physics: (i) How to characterize the statistical properties of sequences of action potentials ("spike trains") produced by neuronal networks ? and; (ii) what are the effects of synaptic plasticity on these statistics ? We introduce a framework in which spike trains are associated to a coding of membrane potential trajectories, and actually, constitute a symbolic coding in important explicit examples (the so-called gIF models). On this basis, we use the thermodynamic formalism from ergodic theory to show how Gibbs distributions are natural probability measures to describe the statistics of spike trains, given the empirical averages of prescribed quantities. As a second result, we show that Gibbs distributions naturally arise when considering "slow" synaptic plasticity rules where the characteristic time for synapse adaptation is quite longer than the characteristic time for neurons dynamics.

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Section: Gibbs Measures As Statistical Modelsmentioning

confidence: 99%

How Gibbs Distributions May Naturally Arise from Synaptic Adaptation Mechanisms. A Model-Based Argumentation

et al. 2009

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“…Using the Riemannian structure of [26] and also [34] it was shown in Section 5 in [36] that the Fisher information is equal to the asymptotic variance (see [42] for the definition). A result taken from [36]:…”

Section: Kl-divergence and Dynamical Information Projectionsmentioning

confidence: 99%

Geodesics and dynamical information projections on the manifold of Hölder equilibrium probabilities

Lopes¹,

Ruggiero²

2022

Preprint

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We consider here the discrete time dynamics described by a transformation T : M → M , where T is either the action of shift T = σ on the symbolic space M = {1, 2, ..., d} N , or, T describes the action of a d to 1 expanding transformation T : S 1 → S 1 of class C 1+α ( for example x → T (x) = d x (mod 1) ), where M = S 1 is the unit circle. It is known that the infinite-dimensional manifold N of equilibrium probabilities for Hölder potentials A : M → R is an analytical manifold and carries a natural Riemannian metric associated with the asymptotic variance. We show here that under the assumption of the existence of a Fourier-like Hilbert basis for the kernel of the Ruelle operator there exists geodesics paths. When T = σ and M = {0, 1} N such basis exists.In a different direction, we also consider the KL-divergence D KL (µ 1 , µ 2 ) for a pair of equilibrium probabilities. If D KL (µ 1 , µ 2 ) = 0, then µ 1 = µ 2 . Although D KL is not a metric in N , it describes the proximity between µ 1 and µ 2 . A natural problem is: for a fixed probability µ 1 ∈ N consider the probability µ 2 in a convex set of probabilities in N which minimizes D KL (µ 1 , µ 2 ). This minimization problem is a dynamical version of the main issues considered in information projections. We consider this problem in N , a case where all probabilities are dynamically invariant, getting explicit equations for the solution sought. Triangle and Pythagorean inequalities will be investigated.

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“…In other words, we would like to estimate a Thermodynamic Formalism version of (1.24) in [6]. (44) is related to the Fisher Information which is a quite important concept in Statistics, Information Geometry and Statistical Mechanics (see section 7.5 in [29], Section 7 in [21], Section 1.6.4 in [8], [54], [55], [33] or [28]. The parameter θ can be considered as time in a weakly relaxing setting of non equilibrium (see expression (29) in section E in [2]).…”

Section: Information Geometry For Gibbs Measuresmentioning

confidence: 99%

“…The results in [33] about Fisher information are related to the asymptotic efficiency of maximum likelihood estimators (see section 4 in [33]).…”

Section: Information Geometry For Gibbs Measuresmentioning

confidence: 99%

Nonequilibrium in Thermodynamic Formalism: the Second Law, gases and Information Geometry

Lopes¹,

Ruggiero²

2021

Preprint

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In Nonequilibrium Thermodynamics and Information Theory, the relative entropy (or, KL divergence) plays a very important role. Consider a Hölder Jacobian J and the Ruelle (transfer) operator L log J . Two equilibrium probabilities µ 1 and µ 2 , can interact via a discretetime Thermodynamic Operation given by the action of the dual of the Ruelle operator L * log J . We argue that the law µ → L * log J (µ), producing nonequilibrium, can be seen as a Thermodynamic Operation after showing that it's a manifestation of the Second Law of Thermodynamics. We also show that the change of relative entropy satisfiesFurthermore, we describe sufficient conditions on J, µ 1 for getting h(L * log J (µ 1 )) ≥ h(µ 1 ), where h is entropy. Recalling a natural Riemannian metric in the Banach manifold of Hölder equilibrium probabilities we exhibit the second-order Taylor formula for an infinitesimal tangent change of KL divergence; a crucial estimate in Information Geometry. We introduce concepts like heat, work, volume, pressure, and internal energy, which play here the role of the analogous ones in Thermodynamics of gases. We briefly describe the MaxEnt method.

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Estimating functionals of one-dimensional Gibbs states

Cited by 9 publications

References 7 publications

How Gibbs Distributions May Naturally Arise from Synaptic Adaptation Mechanisms. A Model-Based Argumentation

How Gibbs Distributions May Naturally Arise from Synaptic Adaptation Mechanisms. A Model-Based Argumentation

Geodesics and dynamical information projections on the manifold of Hölder equilibrium probabilities

Nonequilibrium in Thermodynamic Formalism: the Second Law, gases and Information Geometry

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